## 1491 Reputation

8 years, 281 days

## Seq....

```w := proc (x, y) options operator, arrow; piecewise(y <= .5, -2*tanh(y-.25), .5 < y, 2*tanh(.75-y)) end proc;
Matrix([seq([seq(w(x, y), x = 0 .. 10)], y = 0 .. 10)]);

#or;

Matrix([seq([seq(w(x, y), y = 0 .. 10)], x = 0 .. 10)]);
```

## A numeric method....

You don't gives values of constans,so I assuming.

```restart;
Digits := 20;
Theta := (1/3)*Pi; Upsilon := 1/10;#assume!
eq := ((D@@2)(u))(r) = (-(D(u))(r)^2*u(r)+((Upsilon-1)*(1/2))*(1-u(r)^2-(D(u))(r)^2)*((D(u))(r)*cot(Theta)+2*u(r)))/((D(u))(r)^2-((Upsilon-1)*(1/2))*(1-u(r)^2-(D(u))(r)^2));
sol := dsolve({eq, u(13.75) = .7787, (D(u))(13.75) = .344037}, numeric, abserr = 1.*10^(-16));
plots:-odeplot(sol, [[r, u(r)], [r, (D(u))(r)]], r = 0 .. 13.75, legend = [typeset("Curve: ", u(r)), typeset("Curve: ", (D(u))(r))]);```

## Bug!...

It's possible Bug in Ode Analyzer Assistant.

You must write equation like so:

```y[1](x) -> y1(x)
y[2](x) -> y2(x)```
```diff(y1(x), x\$2)+3*(diff(y1(x), x\$1))+2*y1(x)+2*y2(x) = 3,

diff(y2(x), x\$2)+diff(y2(x), x\$1)+2*y2(x)+2*y1(x) = 8;```

Regards Mariusz.

## ....

maybe like so:

```expand(eval(pdsolve({diff(u(x, z, t), z)+C = 0, u(x, a, t) = 0}), a = h(x, t)));

#u(x, z, t) = C*h(x, t)-C*z```

## ....

Maple has this function Totient(n) built-in.This command was introduced in Maple 2016. Using showstat function we can preview the code.

```with(NumberTheory):
showstat(Totient);
showstat(PrimeFactors);```
```NumberTheory:-Totient := proc(n::{posint, And(algebraic,Not({boolean, `in`, complexcons, extended_numeric}))}, \$)
local prime_factors, p;
1   if not type(n,'posint') then
2       return ('procname')(n)
elif isprime(n) then
3       return n-1
else
4       prime_factors := NumberTheory:-PrimeFactors(n);
5       return n*mul(p-1,`in`(p,prime_factors))/convert(prime_factors,'`*`')
end if
end proc

NumberTheory:-PrimeFactors := proc(n::{integer, And(algebraic,Not({boolean, `in`, complexcons, extended_numeric}))}, \$)
local i;
1   if n = 0 then
2       error "cannot represent all prime factors of %1", n
elif type(n,{'negint', 'posint'}) then
3       return {seq(i[1],`in`(i,ifactors(n)[2]))}
else
4       return ('procname')(n)
end if
end proc
```

Another code you can find on this webpage: http://oeis.org/A000010 see:MAPLE -> # version 2.

Have fun !

## ....

If yours integral is:

`int(sqrt(ln(x)/x^2-1/x), x)`

then Maple can't find it. Possible it has No closed form.

Mathematica, Rubi,Axiom,SymPy, Maxima  cannot do it, either.

Second one:

`int((ln(x)^(a-1)/x^2-1/x)^(1/2),x)`

the same case.

EDITED:

If You want approximate integral by series see worksheet.

Approximate_indefine_integral_by_Series.mw

## Use convert....

```convert(exp(x), Sum, dummy = n)

or:

convert(exp(x), FormalPowerSeries)```

For Simple function (2 methods):

you can use inverse ztransform:

`Sum(x^n*invztrans(eval(exp(x), x = 1/x), x, n), n = 0 .. infinity)#only works if invztrans can find transfrom.`

or n-th derivative:

`Sum(x^n*(eval(diff(exp(x), x\$n), x = 0))/factorial(n), n = 0 .. infinity)#only works if n-th derivative can find.`

Only another way to calculate Julian date.

Formula from:http://aa.usno.navy.mil/faq/docs/JD_Formula.php

Compute the JD corresponding to 2018 January 11, 18h30m30s UT.

Substituting Y = 2018, M = 1, D = 11,h = 18, m = 30, s = 30;

```restart:
JD := proc (Y, M, D, h, m, s) local x;
x := evalf(367*Y-trunc((7/4)*Y+(7/4)*trunc((1/12)*M+3/4))+trunc((275/9)*M)+D+1721013.5-
(1/2)*signum(100*Y+M-190002.5)+1/2+(1/24)*h+(1/1440)*m+(1/86400)*s); end proc:

JD(2018, 1, 11, 18, 30, 30);

# 2.458130271*10^6
```

## ....

It may not possible to find anti-derivatives(closed form).

https://math.stackexchange.com/questions/1469123/integral-of-ex2

https://math.stackexchange.com/questions/168860/functions-cannot-be-integrated-as-simple-functions?rq=1

Mathematica gives No solution.

An aproximation only can be done e.g by series:

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## ....

Use Maple built function intsolve with method Neumann or with this  paper https://arxiv.org/ftp/arxiv/papers/1309/1309.6311.pdf see another attached file (Fredholm_integral_2ver.)

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_Fredholm_integral_2ver.mw

## Expanding with infinte series...

Only for k=1,4,5,6,7 is real solution.

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## Work for me....

```solve(15-(1/100)*m = 5+(1/600)*m, {m})
#{m = 6000/7}

solve(15-1/(100*m) = 5+1/(600*m), {m})
#{m = 7/6000}```

Regards,Mariusz

Please read in Maple Help (Ctrl+F1 and put "How Do I,Solve an Ordinary Differential Equation?" in Search->Enter):

Scroll down to:

-Solving an ODE Numerically

- Taking Derivatives and Integrals of Numeric Solutions

-  Why You Should Not Use Int or Diff on a Numeric Solution

Corrected file:question-vers_2.mw

(Edited: 2 times) :P

## Try....

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`sol := n > ceil(evalf(solve(product(exp(1/i), i = 1 .. n) = 100, n))); solve(sol, n);`